You may wish to look at
Take
a Square before trying this problem.

You can divide the long diagonal of a square into different
fractions by folding.
 In the first image the second fold joins a corner of the square
to the midpoint of the opposite side.
 In the second image the second fold joins a corner of the
square to a point $\frac{1}{8}$ of the way along the opposite
side.
This problem is about the fractions of the long diagonal of a
square which you can construct in this way.
To start with, we shall only consider points on the side of
the square which can easily be found by folding. That is,
$\frac{1}{2}$s or $\frac{1}{4}$s or $\frac{1}{8}$s and so on.

Investigate the fractions of the long diagonal of a square
that can be created in the way described above. Here are some
examples to think about:
...
...
Can you extend the findings and make generalisations?
Can you justify your generalisations?
What about starting with fractions of the side of the square
that are not so easily found by folding?